Design And Circuit Realization Of Square-Root Topology

Higher-order topological insulator represents a new phase of matter,the characterization of which goes beyond the conventional bulk-boundary correspondence and is attracting significant attention by the broad community.In quantum mechanics,almost all known operators are linear or anti-linear,such as rotation,translation,parity,and time reversal,which allows us to construct a mathematical basis for quantum mechanics based on linear algebra.The nonlinear square root operator is one of the few exceptions.Paul Dirac obtained the epochmaking Dirac equation by applying the squared operation to the Klein-Gordon equation.Inspired by this method,it has been found that novel topologies can be constructed using square-root operation.Recent studies have shown that the circuit system is a concise and powerful platform for testing topological energy band theory,suitable for direct observation of topological states and topological phase transitions.I implemented square-root HOT states characterized by bulk polarization in the honeycomb-kagome hybrid circuit system.The lattice of its parent is obtained by direct summation of the honeycomb sublattice with positive and negative alternating onsite potentials and the kagome sublattice of the breathing type.However,theoretical calculations show that the corner states exist at finite energies at this time and cannot be directly observed by impedance measurements.Without affecting the spatial distribution of the wave function,I introduced a grounded inductor of the same size for each node of the circuit to shift the non-zero energy corner modes to zero energy and thus successfully observed the second-order corner state signal of a square-root higherorder topology.Based on a two-dimensional square-root HOT insulator,I propose a tightbinding model for realizing a square-root higher-order Weyl semimetal by threedimensional vertical stacking of a two-dimensional square HOT insulator and introducing a double-helical interlayer coupling with triple rotational symmetry.The system has both a two-dimensional surface arc state and a one-dimensional hinge state and a quantitative invariant.The body polarization describes its topological features.Then,I designed a three-dimensional stacked honeycomb-kagome topology using an inductor-capacitor circuit network in my circuit experiments and performed impedance and voltage measurements on the bulk,surface,and colorful states,directly observing the Weyl point,the “Fermi arc” surface state,and the hinge state.The experimental results are in agreement with the theoretical calculations.The presence of positive and negative energy exonerations in the open-square higher-order Weyl semimetal in pairs marks its difference from the conventional higher-order Weyl semimetal.In terms of applications,the topological robustness of the hinge and surface states can be used to design electronic devices with high interference immunity.I can also use them for low-power signal transmission,designing topological edgestate-based imaging devices,etc.In addition,topological circuits can be fully integrated by complementary metal-oxide-semiconductor technology,which will hopefully address the outstanding challenges faced by the chip industry.The results of the present paper pave the way for circuit simulations of square-root HOT states and will certainly provide useful inspiration for deeper exploration of topological physics and device applications in other solid-state systems,such as cold atoms,photonic crystals,and elastic media.